Samuel Grushevsky, February 28th, 2014

Shimura subvarieties of the moduli space of polarized abelian varieties are defined

from some number theoretic data. The locus of Jacobians of curves is a geometrically

defined subvariety of the moduli space of principally polarized abelian varieties. Thus

the natural question of describing Shimura subvarieites of the Jacobian locus

intertwines the questions of number theory and algebraic geometry, and in fact it is

expected that in sufficiently high dimension/genus the Jacobian locus contains no

Shimura varieties. In contrast, in genus 4 we construct infinitely many Shimura curves

contained in the Jacobian locus, and in genus 3 we construct infinitely many Shimura

curves contained in the locus of hyperelliptic Jacobians. Based on joint work with

Martin Moeller.